There is a canonical pairing on the Brauer group of a surface over a ﬁnite ﬁeld, and an old conjecture of Tate predicts that this pairing is alternating. In this talk I will present a resolution to Tate’s conjecture. The key new ingredient is a circle of ideas originating in algebraic topology, centered around the Steenrod operations. The talk will advertise these new tools (while assuming minimal background in algebraic topology).